One Billion Seconds

Date: 09/27/98 at 20:54:08
From: Galaxy Girl
Subject: Word problem

I'm trying to figure out how to solve this problem.  

If I was born on 01/02/84, on what day, month, and year will I have 
been alive for a billion seconds?

I already figured out that it would be 31.7 years but I don't 
understand how to get it down to the exact date.

Date: 09/28/98 at 12:58:31
From: Doctor Rick
Subject: Re: Word problem

Hi. You've taken the right first step, dividing by the number of 
seconds in a year, which is 60*60*24*365 = 31,536,000 seconds. 

I want *you* to do your problem, so I will find when *I* turned 
1 billion seconds old. I was born on Sept. 9, 1952. You can work out 
your problem the same way.

We need to figure out approximately when is 31.7 years after 9/9/52, 
so we can find how many leap years there are in that time period. Add 
31.7 to 1952.67 (that's approximately Sept. 9, 2/3 of the way through 
the year), and you get 1984.37, which is roughly May 1984. With a leap 
year every 4 years, there were 8 leap years (1956, 60, 64, 68, 72, 76, 
80, 84). I count 1984 because Feb. 29 is before May, but not 1952 
because Feb. 29 is before September.

Now we can figure exactly how many days there were up to Sept. 9, 1983 
(my birthday before the special day). It was 365*31 + 7 (an extra day 
for each leap year). That makes 11,322 days. (That 8th leap day will 
be used later.)

Let's go back now and figure how many days in a billion seconds. It is
1,000,000,000 / (60*60*24) = 11574.074 days. So on Sept. 9, 1983, I 
had used up 11,322 days and there were 11574.074 - 11322 = 252.074 
days left.

Now we just need to add up the days in each month. We think the 
special day was in May. Let's see ... the days in September through 
March are:

  Sep  Oct  Nov  Dec  Jan  Feb  Mar  Apr
   30 + 31 + 30 + 31 + 31 + 29 + 31 + 30 = 243 days

Notice that I counted a leap day in Feb. Now we're up to April 9 
(remember we started on Sept. 9) We have 252 - 243 = 9 days left. That 
puts the special day on April 18, 1984.

If I were going to do a lot of these calculations, I would find a 
faster way. Some calendars tell you the day of the year, for instance, 
Feb. 2 is the 33rd day of the year. If I found what day Sept. 9 was, 
then added the number of days left and looked up what day that was, it 
would make the last part go faster. But it's good to do it the way I 
did at least once.

Now you do it with your birthdate. Your problem starts a few months 
before mine ended. (And if you're 12, then your own birthday is a year 
or two after that. I hope you'll figure it out your own billion-second 
day, too.)

- Doctor Rick, The Math Forum
  http://mathforum.org/dr.math/